If (x + k) is a factor of f(x), which of the following must be true?
A) x = –k and x = k are roots of f(x)
B) Neither x = –k nor x = k is a root of f(x).
C) f(–k) = 0
D) f(k) = 0
On dividing two numbers, if we are getting remainder equal to zero then the smaller number will be the factor of the larger number.
Answer: Option C f(–k) = 0 is correct.
Let us proceed step by step to determine the correct option.
When we divide any polynomial by (x - a), we obtain a result of the form:
f (x) = (x - a) q (x) + f (a) [ From Euclid's division algorithm ]
If the remainder f (a) = 0, then (x − a) is a factor of f (x) --------(1)
Hence, from the given data in the question,
(x + k) is a factor of f(x)
Therefore, f (-k) = 0 on substituting x = -k [ from equation (1) ]